Proof... = 4

[KDOG]

What.
The.
Fark.

My brain got melted enough talking about Shroedingers cat.

 

[yhelothar]

No, you never run out of pie.

So, is pi always equal to 3.14159....? Or does the value of pi change in different reference frames or under acceleration? Would the value be the same near a large star as it is out in the sticks? Would it be the same measured for a circle "parallel" to a massive star as it would be perpendicular to the star?

But as you mentioned that the length of the sides of the teeth approaches 0 as the number of teeth approach infinity. If the lengths of the sides of the teeth is 0, then circumference = 0 and thus pi=0.

 

[yhelothar]

doesn't work, because you end up with an infinite number of corners on the reduced "circle"

the fundamental geometry is not the same - the reduced circle may *approximate* the real circle in shape, but you still have the corners, no matter how small a reduction you make. the appropriate method to drawing a circle (i suspect) is to make a regualr N-gon where N approaches infinity, and compute the perimeter from that (which should come out to be Pi).

included paint picture: http://pics.bbzzdd.com/users/fenixgoon/pi_proof.png
lets assume this is zoomed in 10000000000000000x, you still have a corner that fails to match the geometry of the circle due to the algorithm used, which yields the extra perimeter.

and .99999... =! 1 according to you?

 

[Fenixgoon]

and .99999... =! 1 according to you?

i'm an engineer, so yes, 0.9999 = 1 😉

but what you have proposed is fundamentally flawed. i suspect it approximates the area of a circle well enough (with some extra since the circle is inscribed) but it most definitely will not give you an accurate value of the perimeter of the circle.

Dr. Pizza would probably provide a more technical/proof-based answer.

 

[mrkun]

A countably infinite set is a set for which there exists a bijection (one-to-one function) between it and the real numbers.

 

[TridenT]

Only simpletons would believe this to be true.

I know it is not true. I'm intelligunt.

 

[Cogman]

and .99999... =! 1 according to you?

where are you getting that? It just so happens that infinity is a very strange thing that does not behave as consistently as you would want. For example, look up the dirac delta function. What is the area of an infinitely thin and infinitely tall thing? 1. (Depending on if the top of the infinity tall thing is jagged or not 😀)

It is the same thing here. In this case, an infinite number of infinite small corners just so happens to add up to a perimeter that is .858407346410207... longer than the actual perimeter of a circle of diameter 1.

Isn't infinity fun?

 

[mpo]

Next you'll tell me that Pythagoras had it wrong and the length of the hypotenuse of a right angle is x + y = z rather than x^2 + y^2 = z^2.

 

[yhelothar]

where are you getting that? It just so happens that infinity is a very strange thing that does not behave as consistently as you would want. For example, look up the dirac delta function. What is the area of an infinitely thin and infinitely tall thing? 1. (Depending on if the top of the infinity tall thing is jagged or not 😀)

It is the same thing here. In this case, an infinite number of infinite small corners just so happens to add up to a perimeter that is .858407346410207... longer than the actual perimeter of a circle of diameter 1.

Isn't infinity fun?

That was in response to this line from Fexigoon:
"lets assume this is zoomed in 10000000000000000x, you still have a corner that fails to match the geometry of the circle due to the algorithm used, which yields the extra perimeter."

This seems analogous to saying:
lets assume this is zoomed in 10000000000000000x, you still have that .000000.....1 that fails to equate to one.

Honestly though, I don't understand the proof of why that OP proof doesn't work. I know the OP proof doesn't work obviously, but the explanation of why it doesn't work is beyond me who doesn't understand all the technicalities used in the explanation.

So because in the algorithm used, infinity is a countable number, it's not a true circle. Only an uncountable infinite number of corners can make up a true circle.

And that would mean that you're counting up irrational numbers instead of rational numbers.

Why would an irrational count of infinite corners make a true circles and not a rational count?

I'm lost. Can some of you math geniuses explain to me?

 

[IronWing]

Next you'll tell me that Pythagoras had it wrong and the length of the hypotenuse of a right angle is x + y = z rather than x^2 + y^2 = z^2.

This is it in a nut shell.

 

[Dumac]

But as you mentioned that the length of the sides of the teeth approaches 0 as the number of teeth approach infinity. If the lengths of the sides of the teeth is 0, then circumference = 0 and thus pi=0.

0*inf != 0
...necessarily

 

[yhelothar]

0*inf != 0
...necessarily

When doing limit problems in calc, 0*inf always = 0. My college math education has failed me.. :|

 

[Dumac]

When doing limit problems in calc, 0*inf always = 0. My college math education has failed me.. :|

http://en.wikipedia.org/wiki/L'Hôpital's_rule

 

[slugg]

Tell me how this sounds:

If P is a point in circle C centered at point Q with radius R, then the distance between P and Q must always equal R. Thus, if we can prove that the distance between center Q and any one point, P', on the circle-like shape determined by the proposed algorithm G is NOT equal to R, then we show that the resulting shape is not a circle.

If P' exists in only algorithm G, but not C, then its distance from Q must be greater than R, since algorithm G only gets asymptotically limited to C, but never equal. Thus, G produces output that is not always part of C, so it's not a circle.

 

[yhelothar]

http://en.wikipedia.org/wiki/L'Hôpital's_rule

It's been a few years since I've learned this stuff, but isn't l'hopitals rule just for 0/0 or inf/inf?
That seems to be what wikipedia says too.

 

[CZroe]

and .99999... =! 1 according to you?

Does a 1ft coiled telephone cord expandable to 4ft equal a 1ft rope? This type of circle has a lot of excess length "kinked" into an approximate area. You can't base circular measurement on it.

 

[yhelothar]

Does a 1ft coiled telephone cord expandable to 4ft equal a 1ft rope? This type of circle has a lot of excess length "kinked" into an approximate area. You can't base circular measurement on it.

That makes sense. I guess the logic works when approximating areas but not arc lengths, hence the logic works with integrals.

 

[Praxis1452]

That makes sense. I guess the logic works when approximating areas but not arc lengths, hence the logic works with integrals.

Just realize that by turning the edges in, the perimeter never changes.

How can a square possible equal a circle in length? It can't, yet by turning in the corners, you never change perimeter. Look at it and it should be obvious this is true.

This method doesn't approximate circumference, it approximates area. Length never actually changes.

 

[silverpig]

No, you never run out of pie.

So, is pi always equal to 3.14159....? Or does the value of pi change in different reference frames or under acceleration? Would the value be the same near a large star as it is out in the sticks? Would it be the same measured for a circle "parallel" to a massive star as it would be perpendicular to the star?

pi = 3.14159... by definition, but C/d varies depending on the curvature of space, and is equal to pi in flat space.

 

[Dumac]

It's been a few years since I've learned this stuff, but isn't l'hopitals rule just for 0/0 or inf/inf?
That seems to be what wikipedia says too.

0/0 is often the same as 0*inf

Think about it

Same goes for inf/inf

 

[preslove]

http://www.youtube.com/watch?v=3sUUZkCWRHo&feature=related

 

[DrPizza]

Let's see. I think someone above meant to say that countably infinite is a one to one bijection to the rational numbers (or counting numbers, or whole numbers; whatever) not the real numbers, for they include the irrational numbers.

the set of positive rational numbers is countable. Here's a simple process:
1/1, 1/2, 1/3, 1/4, 1/5,...
2/1, 2/2, 2/3, 2/4, 2/5,...
3/1, 3/2, 3/3, 3/4, 3/5,...
look at the diagonals: 1/1;1/2, 2/1; 1/3, 2/2, 3/1; 1/4, 2/3, 3/2, 4/1;...
Notice the sum of the numerator and denominators: 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, six 7's, seven 8's, ...
If we order the rational numbers this way we could come up with a formula for all of them based on the counting numbers (1, 2, 3, 4,...)
i.e. I could decide I wanted to know the 1,000,483rd rational number in this order and it would correspond to exactly one rational number. Likewise, you could give me a rational number and I could tell you exactly where it is on the list. To keep the numbers small enough to be able to see how this is, consider 2/7. Since it adds up to 9, it would be term number: 1+2+3+...+7 + 2. (1/1 adds up to 2, two terms add up to 3, three terms add up to 4, ... 7 terms add up to 8, and this would be the 2nd term that adds up to 9 (after the ones that add up to 8.) As a formula, any a/b would be the 1+2+3+...+(a+b-2) +a term. Or, in simpler form, (a+b-1)(a+b-2)/2 + a. i.e. 123/587 would be the
(someone will have to check me; I'm making this stuff up as I go.)
(708)(709)/2+123= 251,109th term.

Now, with the irrational numbers, it doesn't matter how you order them, there will always be more irrational numbers than rational numbers. Infinitely many more, in fact. No matter how you order your numbers in decimal form, I can always create numbers that aren't on your list. Let's write down a bunch of numbers in decimal form. We'll write them down to an infinite number of decimal places. If it's a terminating decimal, we'll just continue it with zeros.

example:
0.1234124312341243...
0.4323422123409823...
0.1512934123981249...
0.1234097812340974...

Now, to construct my number (I believe credit for this goes to Cantor), I'll make it differ from the first number in your list by changing the first decimal place. so, it cannot start with 0.1. It can be 0.2 or 0.3, or... Then, I'll make it so the 2nd decimal place doesn't match your second number. Then, I'll make it so the 3rd decimal place doesn't match your 3rd number. No matter what order you write your numbers down in; no matter what formula you use to match your list to the counting numbers, I'll always be able to come up with a shitload of numbers that aren't on your list. Since I've already demonstrated how the rationals are countable (well, I did the positive rationals. If you want them all, then just start with 1, and do the positives and negatives at the same time; there will be "just as many" of each sign. 0, 1/1, -1/1, 1/2, 2/2, -1/2, -2/2, 1/3, 2/3, 3/3, -1/3, -2/3, -3/3,... If you google, you might even find a better system than the one I made up here) then these new numbers I developed must not be countable.


And lastly - L'Hopital's rule. Not just 0/0 or inf/inf, but it can also be used to find limits of other indeterminate forms such as 1^inf, inf-inf, 0*inf, etc. Of course, when limits are in those forms, some manipulation has to take place to put it in the form of a ratio of the 0/0 or inf/inf type. My particular favorite is when you start taking natural logs to do so. 🙂

 

[mjrpes3]

Can you have irrational irrational numbers? Replace every digit in the irrational number with an irrational number (ir * .1 + ir * .01 + ir * .001...). How many more infinities would that be?

 

[Hacp]

Makes sense until you actually think about it. Then you realize its fucking bullshit.

 

[DivideBYZero]

Haha! AT got Math trolled. Again.